First, I would go to the hardware store and look for locks intended for sliding glass "patio" doors.

Another approach (not very aesthetic) would be to install a hasp and padlock.

The bottom line, however, is that sliding doors inherently are not very secure. There is a lot of information on the internet on how to improve the security, and your hardware store will also have some gadgets for you to install. If you want even moderate security, however, I would replace the slider with a standard hinged door. There will probably be some problems with where the hinged door wants to swing.

I agree with Steve. We had a slider closet door (2 doors 40 inches wide iin total) solid wood. I put a hasp & key lock on it and had a group actually rock the door off the tracks. They didn't get inside but they sure created a mess!!! They were real jerks - they also pried a hasp lock off of our linen closet! Both were clearly marked with signs stating "Private. Owners Closet".

However good your lock is on any internal closet, the fact is that anyone who is in your rental for several days will be able to break in if s/he really wants to get inside. Never store anything there that is precious or irreplaceable to you.

I use a very attractive brass hasp and padlock on bi-fold doors. The hasp crosses over the "fold" part to prevent opening. I also have a cute sign on that says, "locked doors, please don't pull". I have very good guests, so I don't have a problem like some owners have with disrespectful guests.

Personally, I would use the most un-noticable and easily broken into as possible. A few people are simply tempted by the challenge and curiosity of what is inside. If its your underwear and extra toilet paper they will be disappointed. Don't keep liquor in there. If someone really wants to get in, they will. Making it easy will lessen damage.

Depending on your price point and location, many others here have said that they don't lock up the extra TP and supplies and even keep bottles of wine in the cellar (out of sight temptation).

Have a storage closet has a two panel bi-fold hinged on one side. I used the small discrete hasp-type lock below to secure the two separate components doors together in the middle----a confusing explanation at best---but the added feature I found is a clear acrylic sliding piece on the top of the door that discourages BUT does not prevent tugs on the door's knob. I found this little $4 device (and the hasp lock) on Amazon in the 'kid-proof' section--question is, does it come in a size for sliding doors also? Again, this slide only works to discourage curious kids from shaking the door and is not a deterrent to forcefully ripping the door open

Just a suggestion for those that are looking for an alternative to an owners closet. We use a full size file cabinet and keep it next to our washer and dryer. In our master bedroom we have a two drawer that I keep all my tools, painting supplies, hardware etc. This actually works pretty well, be sure and use a cabinet that has locks on them.